Application of fractional order theory of magneto-thermoelasticity to an infinite perfect conducting body with a cylindrical cavity

Abstract

A fractional model of the equations of generalized magneto-thermoelasticity for a perfect conducting isotropic thermoelastic media is given. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. The solution is obtained by a direct approach by using the thermoelastic potential function. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizations, Lord–Shulman theory, and Green–Lindsay theory as well as to the coupled theory.

Appendices

Appendix 1

For the thermal shock problem, we can get the solution by substitution σ _o  = 0 in the Eqs. (58)–(63), we get

$$ \bar{\theta} = \frac{{\theta_{o} }}{\omega}\sum\limits_{i = 1}^{2} {a_{i2} K_{o} (k_{i} \,r)}, $$

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$$ {{\bar{u}}}= - \frac{{{\text{a}}{\theta}_{\text{o}} \,\left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{k}}_{\text{i}} \,{\text{a}}_{\text{i2}}}}{{{\text{k}}_{\text{i}}^{2} - {\zeta}\,{\text{s}}^{2}}}} \,{\text{K}}_{1} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right), $$

(65)

$$ {{\bar{h}}}= - \frac{{{\text{a}}\,{\theta}_{\text{o}} \,\left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{k}}_{\text{i}}^{2} {\text{a}}_{\text{i2}}}}{{{\text{k}}_{\text{i}}^{2} - {\zeta}\,{\text{s}}^{2}}}} \,{\text{K}}_{\text{o}} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right), $$

(66)

$$ {{\bar{E}}}= - \frac{{{\text{a}}\,{\theta}_{\text{o}} \,{\text{s}}\,\left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{k}}_{\text{i}} {\text{a}}_{\text{i2}}}}{{{\text{k}}_{\text{i}}^{2} -\zeta \,{\text{s}}^{2}}}} \,{\text{K}}_{1} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right), $$

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$$ {{\bar{\sigma}}}_{\text{rr}} = \frac{{{\text{a}}\,{\theta}_{\text{o}} \,\left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{a}}_{\text{i2}}}}{{{\text{k}}_{\text{i}}^{2} -\zeta \,{\text{s}}^{2}}}} \left[{\zeta {\beta}^{2} {\text{s}}^{2} {\text{K}}_{\text{o}} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right)+\frac{{{2}\,{\text{k}}_{\text{i}}}}{\text{r}}\,{\text{K}}_{1} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right)} \right], $$

(68)

$$ {{\bar{\sigma}}}_{\psi \psi} = \frac{{{\text{a}}\,{\theta}_{\text{o}} \left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{a}}_{\text{i2}}}}{{{\text{k}}_{\text{i}}^{2} -\zeta \,{\text{s}}^{2}}}} \left[{\left({\zeta {\beta}^{2} {\text{s}}^{2} - 2} \right)\,{\text{K}}_{\text{o}} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right)-\frac{{{2}\,{\text{k}}_{\text{i}}}}{\text{r}}\,{\text{K}}_{1} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right)} \right]. $$

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Appendix 2

For the mechanical shock problem, we can get the solution by substitution θ _o  = 0 in the Eqs. (58)–(63), we get

$$ {{\bar{\theta}}}=\frac{{{\sigma}_{\text{o}}}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {{\text{a}}_{\text{i1}} \,{\text{K}}_{\text{o}} ( {\text{k}}_{\text{i}} \,{\text{r)}}}, $$

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$$ {{\bar{u}}}= - \frac{{{\text{a}}{\sigma}_{\text{o}} \,\left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{k}}_{\text{i}} {\text{a}}_{\text{i1}} }}{{{\text{k}}_{\text{i}}^{2} - \zeta \,{\text{s}}^{2}}}} \,{\text{K}}_{1} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right). $$

(71)

$$ {{\bar{h}}}= - \frac{{{\text{a}}{\sigma}_{\text{o}} \left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{k}}_{\text{i}}^{2} \,{\text{a}}_{\text{i1}}}}{{{\text{k}}_{\text{i}}^{2} - \zeta \,{\text{s}}^{2}}}} \,{\text{K}}_{\text{o}} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right), $$

(72)

$$ {{\bar{E}}}= - \frac{{{\text{a}}\,{\text{s}}\,{\sigma}_{\text{o}} \,\left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{k}}_{\text{i}} {\text{a}}_{\text{i1}} }}{{{\text{k}}_{\text{i}}^{2} - \zeta \,{\text{s}}^{2}}}} \,{\text{K}}_{1} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right), $$

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$$ {{\bar{\sigma}}}_{\text{rr}} = \frac{{{\text{a}}\,{\sigma}_{\text{o}} \left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{a}}_{\text{i1}}}}{{{\text{k}}_{\text{i}}^{2} - \zeta \,{\text{s}}^{2}}}} \left[{\zeta \,{\beta}^{2} \,{\text{s}}^{2} {\text{K}}_{\text{o}} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right)+\frac{{{2}\,{\text{k}}_{\text{i}}}}{\text{r}}\,{\text{K}}_{1} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right)} \right], $$

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$$ {{\bar{\sigma}}}_{\psi \psi} = \frac{{{\text{a}}\,{\sigma}_{\text{o}} \left(1+\frac{{{\upsilon}^{\alpha}}}{\alpha !}{\text{s}}^{\alpha} \right)}}{\omega}\,\sum\limits_{\text{i = 1}}^{2} {\frac{{{\text{a}}_{\text{i1}}}}{{{\text{k}}_{\text{i}}^{2} - \zeta \,{\text{s}}^{2}}}} \left[{\left({\zeta \,{\beta}^{2} \,{\text{s}}^{2} - {2}} \right)\,{\text{K}}_{\text{o}} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right) - \frac{{{2}\,{\text{k}}_{\text{i}}}}{\text{r}}\,{\text{K}}_{1} \left({{\text{k}}_{\text{i}} \,{\text{r}}} \right)} \right]. $$

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